A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d4) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.

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Adaptive techniques have great potential for wide application in enhancing the precision of quantum parameter estimation. We present an adaptive quantum state tomography protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this adaptive quantum state tomography protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our adaptive quantum state tomography protocol can outperform tomography protocols using mutually unbiased bases and the twostage mutually unbiased bases adaptive strategy, even with the simplest product measurements. When nonlocal measurements are available, our adaptive quantum state tomography can beat the Gill-Massar bound for a wide range of quantum states with a modest number of copies. We use only the simplest product measurements to implement two-qubit tomography experiments. In the experiments, we use error-compensation techniques to tackle systematic error due to misalignments and imperfection of wave plates, and achieve about a 100-fold reduction of the systematic error. The experimental results demonstrate that the improvement of adaptive quantum state tomography over nonadaptive tomography is significant for states with a high level of purity. Our results also show that this adaptive tomography method is particularly effective for the reconstruction of maximally entangled states, which are important resources in quantum information.

The precision limit in quantum state tomography is of great interest not only to practical applications but also to foundational studies. However, little is known about this subject in the multiparameter setting even theoretically due to the subtle information trade-off among incompatible observables. In the case of a qubit, the theoretic precision limit was determined by Hayashi as well as Gill and Massar, but attaining the precision limit in experiments has remained a challenging task. Here we report the first experiment that achieves this precision limit in adaptive quantum state tomography on optical polarisation qubits. The two-step adaptive strategy used in our experiment is very easy to implement in practice. Yet it is surprisingly powerful in optimising most figures of merit of practical interest. Our study may have significant implications for multiparameter quantum estimation problems, such as quantum metrology. Meanwhile, it may promote our understanding about the complementarity principle and uncertainty relations from the information theoretic perspective. INTRODUCTIONQuantum state tomography is a procedure for inferring the state of a quantum system from quantum measurements and data processing. [1][2][3] It is a primitive of various quantum information processing tasks such as quantum computation, communication, cryptography and metrology. [4][5][6][7][8] In sharp contrast with the classical world, any measurement on a generic quantum system necessarily induces a disturbance, limiting further attempts to extract information from the system. Therefore, many identically prepared systems are usually required for reliable state determination. Conversely, the precision limit in quantum state tomography offers a perfect window for understanding the distinction between quantum physics and classical physics. [9][10][11][12] Recently, great efforts have been directed to improving the tomographic efficiency given limited quantum resources. 13,14 For example, adaptive measurements have been realised in experiments, which may improve the scaling of the infidelity in certain scenarios. 14,15 However, most studies have been tailored to deal with specific figures of merit under special settings such as pure state or single-parameter models, which admit no easy generalisation to the more challenging and exciting multiparameter estimation problems with general figures of merit. In particular, the tomographic precision limit in the multiparameter setting is still poorly understood; experimental studies are especially rare. To fill this gap, in this work we report the first experiment that achieves the quantum precision limit in adaptive quantum state tomography on optical polarisation qubits.

Quantum resource theories seek to quantify sources of nonclassicality that bestow quantum technologies their operational advantage. Chief among these are studies of quantum correlations and quantum coherence. The former isolates nonclassicality in the correlations between systems, and the latter captures nonclassicality of quantum superpositions within a single physical system. Here, we present a scheme that cyclically interconverts between these resources without loss. The first stage converts coherence present in an input system into correlations with an ancilla. The second stage harnesses these correlations to restore coherence on the input system by measurement of the ancilla. We experimentally demonstrate this interconversion process using linear optics. Our experiment highlights the connection between nonclassicality of correlations and nonclassicality within local quantum systems and provides potential flexibilities in exploiting one resource to perform tasks normally associated with the other.

Collective measurements on identically prepared quantum systems can extract more information than local measurements, thereby enhancing information-processing efficiency. Although this nonclassical phenomenon has been known for two decades, it has remained a challenging task to demonstrate the advantage of collective measurements in experiments. Here, we introduce a general recipe for performing deterministic collective measurements on two identically prepared qubits based on quantum walks. Using photonic quantum walks, we realize experimentally an optimized collective measurement with fidelity 0.9946 without post selection. As an application, we achieve the highest tomographic efficiency in qubit state tomography to date. Our work offers an effective recipe for beating the precision limit of local measurements in quantum state tomography and metrology. In addition, our study opens an avenue for harvesting the power of collective measurements in quantum information-processing and for exploring the intriguing physics behind this power.

Full quantum state tomography (FQST) plays a unique role in the estimation of the state of a quantum system without a priori knowledge or assumptions. Unfortunately, since FQST requires informationally (over)complete measurements, both the number of measurement bases and the computational complexity of data processing suffer an exponential growth with the size of the quantum system. A 14-qubit entangled state has already been experimentally prepared in an ion trap, and the data processing capability for FQST of a 14-qubit state seems to be far away from practical applications. In this paper, the computational capability of FQST is pushed forward to reconstruct a 14-qubit state with a run time of only 3.35 hours using the linear regression estimation (LRE) algorithm, even when informationally overcomplete Pauli measurements are employed. The computational complexity of the LRE algorithm is first reduced from ∼10 19 to ∼10 15 for a 14-qubit state, by dropping all the zero elements, and its computational efficiency is further sped up by fully exploiting the parallelism of the LRE algorithm with parallel Graphic Processing Unit (GPU) programming. Our result demonstrates the effectiveness of using parallel computation to speed up the postprocessing for FQST, and can play an important role in quantum information technologies with large quantum systems.

Quantum coherence, which quantifies the superposition properties of a quantum state, plays an indispensable role in quantum resource theory. A recent theoretical work [Phys. Rev. Lett. 116, 070402 (2016)] studied the manipulation of quantum coherence in bipartite or multipartite systems under the protocol Local Incoherent Operation and Classical Communication (LQICC). Here we present the first experimental realization of obtaining maximal coherence in assisted distillation protocol based on linear optical system. The results of our work show that the optimal distillable coherence rate can be reached even in one-copy scenario when the overall bipartite qubit state is pure. Moreover, the experiments for mixed states showed that distillable coherence can be increased with less demand than entanglement distillation. Our work might be helpful in the remote quantum information processing and quantum control.Quantum coherence, which exhibits the fundamental signature of superposition in quantum mechanics, has been exploited in many fields of quantum physics, such as biological systems [1][2][3], transport theory [4,5], thermodynamics [6][7][8][9][10][11][12], nanoscale physics [13] and other scientific work associated with quantum theory [14][15][16][17][18][19][20][21]. Recently, rigorous and unambiguous framework for quantifying quantum coherence has also been put forward, which has enhanced the exploitation of its operational significance in the context of quantum resource theory [22][23][24].Quantifying quantum coherence starts from the definition of incoherent states (free states) and incoherent operations (free operations) [22,23,25]. A quantum state ρ is incoherent if it is diagonal in a given reference basis {|i }, i.e., ρ = i p i |i i|, with some probability {p i } [23,26]. Incoherent operators are required to fullfillK n IK † n ⊂ I for all n represented by the set of Kraus operators {K n }, where I is the set of incoherent states. Moreover, in a d-dimensional Hilbert space H, the maximally coherent state is |Φ d = 1/d i |i , and |Φ := |Φ 2 denotes the unit coherence resource state [22].As a quantification and measure of quantum superposition in a fixed basis {|i }, various coherence measures have been proposed [22,24,26,27]. In this paper, we choose relative entropy of coherence [22] as the measure of this property. Relative entropy of coherence of a quantum state ρ is given by C r (ρ) = S(∆(ρ))− S(ρ), where ∆(ρ) = Σ i |i i| ρ |i i| is the dephasing in the reference basis. In many recent works, this kind of measure has been endowed with special significance as well as operational meaning [22][23][24]. Winter and Yang [23] showed that asymptotically the standard distillable coherence of a general quantum state is given by the * These authors contributed equally to this work.† gyxiang@ustc.edu.cn relative entropy of quantum coherence. Likewise, Ma's group [24] has shown that the intrinsic randomness as a measure of coherence is just equal to the relative entropy of quantum coherence. Recently, the manipulation...